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JHEP06(2012)070

Published for SISSA bySpringer

Received: March 16, 2012

Accepted: May 24, 2012

Published: June 12, 2012

One-loop BPS amplitudes as BPS-state sums

Carlo Angelantonj,a,bIoannis Florakisc,dand Boris Piolineb,e

aDipartimento di Fisica, Universit` a di Torino, and INFN Sezione di Torino,

Via P. Giuria 1, 10125 Torino, Italy

bDep PH-TH, CERN,

1211 Geneva 23, Switzerland

cArnold Sommerfeld Center for Theoretical Physics, Fakult¨ at f¨ ur Physik,

Ludwig-Maximilians-Universit¨ at M¨ unchen,

Theresienstr. 37, 80333 M¨ unchen, Germany

dMax-Planck-Institut f¨ ur Physik, Werner-Heisenberg-Institut,

80805 M¨ unchen, Germany

eLaboratoire de Physique Th´ eorique et Hautes Energies, CNRS UMR 7589,

Universit´ e Pierre et Marie Curie – Paris 6,

4 place Jussieu, 75252 Paris cedex 05, France

E-mail: carlo.angelantonj@unito.it, florakis@mppmu.mpg.de,

boris.pioline@cern.ch

Abstract: Recently, we introduced a new procedure for computing a class of one-loop

BPS-saturated amplitudes in String Theory, which expresses them as a sum of one-loop

contributions of all perturbative BPS states in a manifestly T-duality invariant fashion.

In this paper, we extend this procedure to all BPS-saturated amplitudes of the form

?

convergent Poincar´ e series, against which the fundamental domain F can be unfolded. The

resulting BPS-state sum neatly exhibits the singularities of the amplitude at points of gauge

symmetry enhancement, in a chamber-independent fashion. We illustrate our method with

concrete examples of interest in heterotic string compactifications.

FΓd+k,dΦ, with Φ being a weak (almost) holomorphic modular form of weight −k/2. We

use the fact that any such Φ can be expressed as a linear combination of certain absolutely

Keywords: Superstrings and Heterotic Strings, String Duality

ArXiv ePrint: 1203.0566

Open Access

doi:10.1007/JHEP06(2012)070

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JHEP06(2012)070

Contents

1 Introduction1

2Niebur-Poincar´ e series and almost holomorphic modular forms

2.1Various Poincar´ e series

2.2Fourier expansion of the Niebur-Poincar´ e series

2.3 Harmonic Maass forms from Niebur-Poincar´ e series

2.4 Weak holomorphic modular forms from Niebur-Poincar´ e series

2.5 Weak almost holomorphic modular forms from Niebur-Poincar´ e series

2.6Summary

4

4

7

8

11

13

14

3 A new road to one-loop modular integrals

3.1 Niebur-Poincar´ e series in a nutshell

3.2One-loop BPS amplitudes as BPS-state sums

3.3 One-loop BPS amplitudes with momentum insertions

3.4 BPS-state sum for integer s

3.5Singularities at points of gauge symmetry enhancement

16

16

17

22

22

25

4 Some examples from string threshold computations

4.1 A gravitational coupling in maximally supersymmetric heterotic vacua

4.2 Gauge-thresholds in N = 2 heterotic vacua with/without Wilson lines

4.3K¨ ahler metric corrections in N = 2 heterotic vacua

4.4 An example from non-compact heterotic vacua

26

26

27

28

29

A Notations and useful identities

A.1 Operators acting on modular forms

A.2 Whittaker and hypergeometric functions

A.3 Kloosterman-Selberg zeta function

29

30

31

34

B Selberg-Poincar´ e series vs. Niebur-Poincar´ e series 35

1Introduction

Scattering amplitudes in closed string theory involve, at h-th order in perturbation theory,

an integral over the moduli space of conformal structures on genus h closed Riemann

surfaces. The torus amplitude (corresponding to h = 1) is particularly relevant, as it

encodes the perturbative spectrum of excitations. Moreover, for special choices of vacua

and of external states, corresponding to a special class of F-term interactions in the low

energy effective action, the torus contribution exhausts the perturbative series, and thus can

serve as a basis for quantitative tests of string dualities (see e.g. [1] and references therein).

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JHEP06(2012)070

The moduli space of conformal metrics on the torus is the Poincar´ e upper half plane H,

parameterised by the complex structure parameter τ = τ1+ iτ2, modulo the action of the

modular group SL(2,Z). After performing the path integral over the world-sheet fields and

over the location of the vertex-operator insertions, the relevant amplitude is then expressed

as a modular integral

?

where F = {τ ∈ H| −1

τ−2

2

dτ1dτ2 is the SL(2,Z)-invariant integration measure, and A is a modular-invariant

function whose precise expression depends on the problem at hand. With this choice of

integration domain, the imaginary part τ2can be identified with Schwinger’s proper time,

while the real part τ1 is the Lagrange multiplier imposing the level-matching condition.

Part of the difficulty in evaluating integrals of the form (1.1) is the unwieldy shape of F,

which intertwines the integrals over τ1and τ2.

Depending on the function A(τ1,τ2) methods have been devised to overcome this

problem. If A is a weak almost holomorphic function1of τ (or, alternatively, an anti-

holomorphic function), the surface integral over F can be reduced by Stokes’ theorem to a

line-integral over its boundary ∂F that can be explicitly computed [2]. On the contrary, if

A is a genuine non-holomorphic function, as is the case for the one-loop partition function

of closed-oriented strings, no useful method is known to evaluate the integral, but one can

use the Rankin-Selberg-Zagier transform [3] to connect the integral to the graded sum of

physical degrees of freedom [4–7]. A frequently encountered intermediate case is that of

modular integrals of the form

?

where

Γd+k,d(G,B,Y ;τ1,τ2) ≡ τd/2

F

dµA(τ1,τ2), (1.1)

2≤ τ1<1

2,|τ| ≥ 1} is the standard fundamental domain, dµ =

F

dµΓd+k,d(G,B,Y ;τ1,τ2)Φ(τ), (1.2)

2

?

pL,pR

q

1

4p2

L¯ q

1

4p2

R

(1.3)

is the partition function of the Narain lattice of Lorentzian signature (d + k,d), G, B, Y

parameterise the Narain moduli space SO(d+k,d)/SO(d+k)×SO(d), and Φ(τ) is a weak

almost holomorphic modular form of negative weight w = −k/2, which we shall refer to as

the elliptic genus. Such integrals occur in particular in one-loop corrections to certain BPS-

saturated couplings in the low energy effective action of heterotic or type II superstrings.

The traditional approach in the physics literature for computing modular integrals

of the form (1.2) has been the orbit method, which proceeds by unfolding the integration

domain F against the lattice partition function Γd+k,d[8–19]. While this procedure yields

an infinite series expansion which is useful in certain limits in Narain moduli space, it does

1By weak almost holomorphic we mean an element in the graded polynomial ring generated by the

holomorphic Eisenstein series E4 and E6, the almost holomorphic Eisenstein seriesˆE2 and the inverse

of the discriminant 1/∆. Our notations for Eisenstein series and other modular forms are collected in

appendix A.1. The adverb weak refers to the fact that the only singularity is, at most, a finite order pole

at the cusp q = 0.

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not make manifest the invariance under the T-duality group O(d + k,d,Z) of the Narain

lattice, nor does it clearly display the singularities of the amplitude at points of gauge

symmetry enhancement.

In [20] we proposed a new method for dealing with modular integrals of the form (1.2),

which relies on representing the elliptic genus Φ as a Poincar´ e series, and on unfolding

the integration domain against it rather than against the lattice partition function. The

advantage of this method is that T-duality remains manifest at all steps, and the result is

valid in all chambers in Narain moduli space, unlike the conventional approach.2Moreover,

the amplitude is expressed as a sum over all BPS states in the spectrum, thus generalising

the constrained Eisenstein series constructed in [21].3

amplitude at points of enhanced gauge symmetry can be immediately read-off from the

contributions of those BPS states which become massless.

The main difficulty in implementing this strategy is due to the fact that the standard

Poincar´ e series representation of a weak holomorphic modular form of weight w ≤ 0 [28–30]

is only conditionally convergent, and therefore unsuited for unfolding. In [20] we attempted

to circumvent this problem by considering a class of non-holomorphic Poincar´ e series

E(s,κ,w) that provide a natural regularisation of the modular forms of interest by inserting

a Kronecker-type convergence factor τs−w/2

2

in the standard sum over images. Therefore,

the resulting Poincar´ e series, originally studied in [31], converges absolutely for ?(s) > 1,

and the modular integral?

w

2, where E(s,κ,w) becomes formally a holomorphic function of τ. This procedure would

then allow to compute the modular integral (1.2) for any Φ which can be expressed as a

linear combination of such E(w

2,κ,w)’s, at least in principle. However, this strategy turned

out to be quite difficult in practice, since this analytic continuation depends on the notori-

ously subtle analytic properties of the Kloosterman-Selberg zeta function which appears in

the Fourier expansion of E(s,κ,w). That is the reason why the analysis [20] was restricted

to the case of zero modular weight, where the analytic continuation is fully under control.

In the present work, we overcome these difficulties by employing a different class of

non-analytic Poincar´ e series introduced in the mathematics literature by Niebur [32] and

Hejhal [33] and studied more recently by Bruinier, Ono and Bringmann [34–37]. Similarly

to the Selberg-Poincar´ e series E(s,κ,w), the Niebur-Poincar´ e series F(s,κ,w) converges

absolutely for ?(s) > 1, and formally becomes holomorphic in τ at the point s =

However, the Niebur-Poincar´ e series can be specialised to the other interesting value s =

1−w

Although at this value F(s,κ,w) belongs to the more general class of weak harmonic Maass

forms,4that are typically non-holomorphic functions of τ, it has the important property

Finally, the singularities of the

FΓd,dE(s,κ,w) can be computed by unfolding F against it, at

least for large s. The result should then be analytically continued to the desired value s =

w

2.

2, which lies inside the domain of absolute convergence when the weight w is negative.

2See for instance [18] for a detailed discussion on chamber dependence of the traditional unfolding

method.

3BPS states sums have appeared in earlier works [22–26]. In our approach these BPS sums follow directly

from unfolding the fundamental domain against the elliptic genus, without any further assumption.

4A harmonic Maass form is an eigenmode of the weight-w Laplacian on H with the same eigenvalue as

weak holomorphic modular forms. The positive frequency part of a weak harmonic Maass form is sometimes

known as a Mock modular form. See section 2.3 for a more precise definition of weak harmonic Maass forms.

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that any linear combination of F(1 −w

principal part of a given weak holomorphic modular form Φ, is actually a weak holomorphic

modular form, and equals Φ itself. Therefore, given any weak holomorphic modular form

Φ, the integral (1.2) can be computed by decomposing Φ into a sum of Niebur-Poincar´ e

series, and by unfolding each of them against the integration domain. Moreover, the same

strategy works also for weak almost holomorphic modular forms (i.e. involving powers of

ˆE2), where now one has to specialise the Niebur-Poincar´ e series to the values s = 1−w

with n a non-negative integer.

The outline of this work is as follows. In section 2, we introduce the Niebur-Poincar´ e

series F(s,κ,w), discuss their main properties, present their Fourier coefficients and identify

their limiting values at s = 1 −w

result that any weak almost holomorphic modular form can be represented as a linear

combination of them. In section 3 we evaluate the modular integral?

we use this result to compute a sample of modular integrals of physical interest of the

form (1.2). In appendix A, we define our notation for modular forms, we collect various

definitions and properties of Whittaker and hypergeometric functions, and we introduce the

Kloosterman sums and the Kloosterman-Selberg zeta function. Finally, in appendix B we

briefly discuss the relation between the Selberg- and Niebur-Poincar´ e series, and between

the “shifted constrained” Epstein zeta series and the above BPS-state sums. The reader

interested only in physics applications may skip section 2 and proceed directly to section 3,

which begins with an executive summary of the main properties of F(s,κ,w).

2,κ,w), whose coefficients are determined by the

2+n,

2+ n. We conclude the section by showing the important

FΓd+k,dF?s,κ,−k

2

?

in terms of certain BPS-state sums and discuss their singularity structure. In section 4,

Note added.

of ref. [34] where similar computations have been performed for general even lattices of

signature (d + k,d) with d = 0,1,2, in particular reproducing Borcherds’ automorphic

products for d = 2 [38]. Unlike [34], we restrict the analysis to even self-dual lattices

(with k = 0 mod 8), which suffices for our physics applications, but we allow for almost

holomorphic modular forms and arbitrary dimension d.

After having obtained most of the results in this paper, we became aware

2 Niebur-Poincar´ e series and almost holomorphic modular forms

In this section, we introduce the Niebur-Poincar´ e series F(s,κ,w), a modular invariant

regularisation of the na¨ ıve Poincar´ e series of negative weight. We present its Fourier ex-

pansion for general values of s, and analyse its limit as s → 1 −w

non-negative integer. We explain how to represent any weak almost holomorphic modular

form of negative weight as a suitable linear combinations of such Poincar´ e series.

2+ n where n is any

2.1 Various Poincar´ e series

In order to motivate the construction of the Niebur-Poincar´ e series, let us start with a brief

overview of Poincar´ e series in general. Let w be an even integer5and f a function on the

5In this paper we shall restrict to the case of even weight w in order to avoid complications with non-

trivial multiplier systems, though the construction can be generalised to half-integer weights.

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